Vague Soft Expert Set and its Application in Decision...

Malaysian Journal of Mathematical Sciences 11(1): 23 – 39 (2017)

MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES

Journal homepage: http://einspem.upm.edu.my/journal

Vague Soft Expert Set and its Application inDecision Making

Khaleed Alhazaymeh 1 and Nasruddin Hassan∗2

1Department of Basic Science and Mathematics, PhiladelphiaUniversity, Jordan

2School of Mathematical Sciences, Faculty of Science andTechnology, Universiti Kebangsaan Malaysia, Malaysia

E-mail: [email protected]∗Corresponding author

Received: 23rd May 2016Accepted: 18th November 2016

ABSTRACT

In this paper, we recall the concept of vague soft expert set theory and itsoperations. We define the AND and OR operations of vague soft expertset and some properties. We prove the De Morgan’s law on vague softexpert set. We then provide an illustrative example of vague soft expertset and its application in decision making.

Keywords: Fuzzy soft expert set theory, vague set, vague soft set, vaguesoft expert set.

Khaleed Alhazaymeh and Nasruddin Hassan

1. Introduction

Fuzzy set was introduced by Zadeh (1965) as a mathematical tool to solveproblems and vagueness in everyday life. It has been currently extended byVarnamkhasti and Hassan (2012, 2015) to neurogenetics and neuro-fuzzy infer-ence (Varnamkhasti and Hassan (2013)). Molodtsov (1999) mentioned a softset as a mathematical way to represent and solve these problems with uncer-tainties which traditional mathematical tools cannot handle. He has provedseveral applications of his theory in solving problems in economics, engineer-ing, environment, social science, medical science and business management.Roy and Maji (2007) used this theory to solve some decision-making prob-lems, thus extending decision-making beyond goal programming (Hassan andAyop (2012), Hassan and Halim (2012), Hassan and Loon (2012), Hassan andSahrin (2012), Hassan et al. (2012)). Furthermore, soft set has been developedrapidly to multiparametrized soft set (Alkhazaleh et al. (2011), Salleh et al.(2012)), soft intuitionistic fuzzy sets (Alhazaymeh et al. (2012)), probabilitytheory (Zhu and Wen (2010)), trapezoidal fuzzy soft sets (Khalil and Hassan(2016, 2017)), Q-fuzzy soft sets by Adam and Hassan (2014b,e), followed byQ-fuzzy soft matrix by Adam and Hassan (2014c,d) and multi Q-fuzzy soft setsin Adam and Hassan (2014a, 2015, 2016a,b), while soft expert sets was pro-posed by Alkhazaleh and Salleh (2011). Vague set in Gau and Buehrer (1993)and vague soft set in Xu et al. (2010) was applied by Alhazaymeh and Has-san (2012a,b,c) to decision making, and further explored interval-valued vaguesoft set (Alhazaymeh and Hassan (2013a,c)), generalized vague soft expert set(Alhazaymeh and Hassan (2013b, 2014a,b,c)), and vague soft set relations andfunctions (Alhazaymeh and Hassan (2015)), and proceeded by Alzu’bi et al.(2015) and Tahat et al. (2015).

In this paper, we review and extend the concept of vague soft expert settheory (Hassan and Alhazaymeh (2013)) and soft expert sets (Bashir and Salleh(2012), Alkhazaleh and Salleh (2014), Sahin et al. (2015), Selvachandran andSalleh (2015), Al-Quran and Hassan (2016), Qayyum et al. (2016)) to furtherillustrate an application of this theory in decision making. The concept of vaguesoft expert set is explored as a more effective and useful mathematical tool,which is a combination between a vague set and a soft expert set, where the usercan know the opinion of all experts in one model with a single operation. Theorganization of this paper is as follows: In Section 2 basic notions about vaguesoft expert set is reviewed. Section 3 discusses the AND and OR operationson vague soft expert set. We also study some De Morgan’s law based on theseoperations. In Section 4 we discuss the propositions on vague soft expert set andsome examples are elaborated. In Section 6 we present the application basedon vague soft expert set. The last section summarizes all the contributions andpoints out future research work.

24 Malaysian Journal of Mathematical Sciences

Vague Soft Expert Set and its Application in Decision Making

2. Preliminaries

In this section, we recall some basic notions related to vague soft expert settheory.

Let U be a universal, E be a set of parameters, X a set of expert (agents),and O = {1 = agree, 0 = disagree} a set of opinions. Let Z = E ×X ×O andA ⊂ Z.

Definition 2.1. Alkhazaleh and Salleh (2011). A pair (F,A) is called a softexpert set over U , where F is a mapping given by

F : A→ P (U)

where P (U) denotes the power set of U .

Definition 2.2. (Alkhazaleh and Salleh (2011)). An agree-soft expert set(F,A)1 over U is a soft expert subset from (F,A) defined as follows:

(F,A)1 = {F1(α) : α ∈ E ×X × {1}}.

Definition 2.3. (Alkhazaleh and Salleh (2011)). A disagree-soft expert set(F,A)0 over U is a soft expert subset from (F,A) defined as follows:

(F,A)0 = {F0(α) : α ∈ E ×X × {0}}.

Definition 2.4. (Hassan and Alhazaymeh (2013)). A pair (F,A) is calledvague soft expert set over U , where F is a mapping given by

F : A −→ V U

where V U denotes the set of all vague subsets of U .

Definition 2.5. (Hassan and Alhazaymeh (2013)). For two vague soft expertsets (F,A) and (G,B) over U , (F,A) is called a vague soft expert subset of(G,B) ifa. A ⊆ B,b. ∀ε ∈ A,F (ε) is a vague subset of G(ε).

This relationship is denoted by (F,A) ⊆ (G,B). In this case (G,B) is called avague soft expert superset of (F,A).

Definition 2.6. (Hassan and Alhazaymeh (2013)). Two vague soft expert sets(F,A) and (G,B) over U are said to be equal if (F,A) is a vague soft expertsubset of (G,B) and (G,B) is a vague soft expert subset of (F,A).

Definition 2.7. (Hassan and Alhazaymeh (2013)). An agree-vague soft expertset (F,A)1 over U is a vague soft expert subset of (F,A) defined as follows:

(F,A)1 = {F1(α) : α ∈ E ×X × {1}}.

Malaysian Journal of Mathematical Sciences 25


Definition 2.8. (Hassan and Alhazaymeh (2013)). A disagree-vague soft ex-pert set (F,A)0 over U is a vague soft expert subset of (F,A) defined as follows:

(F,A)0 = {F0(α) : α ∈ E ×X × {0}}.

Definition 2.9. (Hassan and Alhazaymeh (2013)). The complement of a vaguesoft expert set (F,A) is denoted by (F,A)c and is defined by (F,A)c = (F c, eA)where F c : A −→ V U is a mapping given by

F c(α) = c(F (α)),∀α ∈ A,

where c is a vague complement.

Definition 2.10. (Hassan and Alhazaymeh (2013)). The union of two vaguesoft expert sets (F,A) and (G,B) over U , denoted by (F,A)∪(G,B), is thevague soft expert set (H,C) where C = A ∪B and ∀ε ∈ C,

H(ε) =

F (ε) , ifε ∈ A−BG(ε) , ifε ∈ B −A

max (F (ε), G(ε)) , ifε ∈ A ∩B

Definition 2.11. (Hassan and Alhazaymeh (2013)). The intersection of twovague soft expert sets (F,A) and (G,B) over U , denoted by (F,A)∩(G,B), isthe vague soft expert set (H,C) where C = A ∩B and ∀ε ∈ C,

H(ε) =

F (ε) , ifε ∈ A−BG(ε) , ifε ∈ B −A

min (F (ε), G(ε)) , ifε ∈ A ∩B

Example 2.1. LetA = {(e1, p, 1), (e2, p, 0), (e3, p, 1), (e1, q, 1), (e2, q, 1), (e1, r, 0), (e2, r, 1), (e3, r, 1)}andB = {(e1p, 1), (e2, p, 0), (e1, q, 1), (e2, q, 1), (e1, r, 0), (e2, r, 1)}.

Suppose (F,A) and (G,B) are two vague soft expert sets over U such that

(F,A) = {((e1, p, 1), { u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉}), ((e1, q, 1), { u1

〈0.3,0.6〉 ,u2

〈0.9,0.9〉 ,u3

〈1,1〉}),((e2, q, 1), { u1

〈0.3,0.4〉 ,u2

〈0.4,0.5〉 ,u3

〈0.6,0.8〉}), ((e2, r, 1), { u1

〈0.1,0.1〉 ,u2

〈0.8,0.9〉 ,u3

〈0.3,0.3〉}),((e3, p, 1), { u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0.5〉}), ((e3, r, 1), { u1

〈0.1,0.2〉 ,u2

〈0.2,0.6〉 ,u3

〈0.4,0.6〉}),((e1, r, 0), { u1

〈0.3,0.7〉 ,u2

〈0.1,0.3〉 ,u3

〈0.3,0.6〉}), ((e2, p, 0), { u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉})}.

(G,B) = {((e1, p, 1), { u1

〈0.3,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉}), ((e1, q, 1), { u1

〈0.4,0.5〉 ,u2

〈0.9,0.9〉 ,u3

〈1,1〉}),((e2, q, 1), { u1

〈0.4,0.4〉 ,u2

〈0.5,0.5〉 ,u3

〈0.7,0.7〉}), ((e2, r, 1), { u1

〈0.1,0.1〉 ,u2

〈0.9,0.9〉 ,u3

〈0.4,0.4〉}),((e1, r, 0), { u1

〈0.4,0.5〉 ,u2

〈0.2,0.2〉 ,u3

〈0.4,0.4〉}), ((e2, p, 0), { u1

〈0.5,0.5〉 ,u2

〈0.7,0.7〉 ,u3

〈0,0〉})}.

By using basic vague union (maximum) we have (F,A) ∪ (G,B) = (H,C)where



(H,C) = {((e1, p, 1), { u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉}), ((e1, q, 1), { u1

〈0.4,0.5〉 ,u2

〈0.9,0.9〉 ,u3

〈1,1〉}),((e2, q, 1), { u1

〈0.4,0.4〉 ,u2

〈0.5,0.5〉 ,u3

〈0.7,0.7〉}), ((e2, r, 1), { u1

〈0.1,0.1〉 ,u2

〈0.9,0.9〉 ,u3

〈0.4,0.4〉}),((e3, p, 1), { u1

〈0.3,0.6〉 ,u2

〈0.6,0.4〉 ,u3

〈0.5,0.5〉}), ((e3, r, 1), { u1

〈0.1,0.2〉 ,u2

〈0.2,0.6〉 ,u3

〈0.4,0.6〉}),((e1, r, 0), { u1

〈0.4,0.5〉 ,u2

〈0.2,0.2〉 ,u3

〈0.4,0.4〉}), ((e2, p, 0), { u1

〈0.5,0.5〉 ,u2

〈0.7,0.7〉 ,u3

〈0,0〉})}.

Example 2.2. Consider Example 2.1 By using basic vague intersection (min-imum) we have (F,A) ∩ (G,B) = (H,C) where

(H,C) = {((e1, p, 1), { u1

〈0.3,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉}), ((e1, q, 1), { u1

〈0.3,0.6〉 ,u2

〈0.9,0.9〉 ,u3

〈1,1〉}),((e2, q, 1), { u1

〈0.3,0.4〉 ,u2

〈0.4,0.5〉 ,u3

〈0.6,0.8〉}), ((e2, r, 1), { u1

〈0.1,0.1〉 ,u2

〈0.5,0.9〉 ,u3

〈0.3,0.4〉}),((e3, p, 1), { u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0.5〉}), ((e3, r, 1), { u1

〈0.1,0.2〉 ,u2

〈0.2,0.6〉 ,u3

〈0.4,0.5〉}),((e1, r, 0), { u1

〈0.3,0.7〉 ,u2

〈0.1,0.3〉 ,u3

〈0.3,0.6〉}), ((e2, p, 0), { u1

〈0.3,0.6〉 ,u2

〈0.6,0.7〉 ,u3

〈0,0.5〉})}.

3. AND and OR operations

In this section we define the AND and OR operations of vague soft expertsets and study their properties using a few examples.

Let U be a universal, E be a set of parameters, X a set of expert (agents),and O = {1 = agree, 0 = disagree} a set of opinions. Let Z = E ×X ×O andA ⊂ Z.

Definition 3.1. If (F,A) and (G,B) are two vague soft expert sets over Uthen “(F,A) AND (G,B)" denoted by (F,A) ∧ (G,B), is defined by

(F,A) ∧ (G,B) = (H,A×B)

where H (α, β) = F (α) ∩ G(β) ∀α, β ∈ A × B and ∩ the vague soft expertintersection.

Example 3.1. Suppose that a company produces new types of products andwants to take the opinion of some experts about these products. Let U ={u1, u2, u3} be a set of products, E = {e1, e2, e3} a set of decision parame-ters where ei = {1, 2, 3} denotes the parameters “easy to use",“quality" and“cheap". Let {p, q, r} be a set of experts. Suppose that

A = {(e1, p, 1) , (e3, p, 1) , (e3, r, 1) , (e2, p, 0)} andB = {(e1, p, 1) , (e3, p, 1) , (e2, p, 0)}.

Suppose (F,A) and (G,B) are two vague soft expert sets over U such that(F,A) ={(

(e1, p, 1) ,{

u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉

}),(

(e3, p, 1) ,{

u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0.5〉

}),(

(e3, r, 1) ,{

u1

〈0.3,0.7〉 ,u2

〈0.1,0.3〉 ,u3

〈0.3,0.6〉

}),(

(e2, p, 0) ,{

u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉

})},



and

(G,B) =

{((e1, p, 1) ,

{u1

〈0.4,0.6〉 ,u2

〈0.7,0.7〉 ,u3

〈1,1〉

}),(

(e3, p, 1) ,{

u1

〈0.4,0.4〉 ,u2

〈0.3,0.7〉 ,u3

〈0.4,0.7〉

}),(

(e2, p, 0) ,{

u1

〈0.4,0.7〉 ,u2

〈0.6,0.7〉 ,u3

〈0,0〉

})}.

Then

(F,A) ∧ (G,B) = (H,A×B) ={(((e1, p, 1) , (e1, p, 1)) ,

{u1

〈0.4,0.6〉 ,u2

〈0.6,0.8〉 ,u3

〈0,1〉

}),

(((e1, p, 1) , (e3, p, 1)) ,

{u1

〈0.4,0.5〉 ,u2

〈0.3,0.8〉 ,u3

〈0,0.7〉

}),(

((e1, p, 1) , (e2, p, 0)) ,{

u1

〈0.4,0.7〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉

}),(

((e3, p, 1) , (e1, p, 1)) ,{

u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,1〉

}),(

((e3, p, 1) , (e3, p, 1)) ,{

u1

〈0.3,0.6〉 ,u2

〈0.3,0.7〉 ,u3

〈0.4,0.7〉

}),(

((e3, p, 1) , (e2, p, 0)) ,{

u1

〈0.3,0.7〉 ,u2

〈0.6,0.7〉 ,u3

〈0,0.5〉

}),(

((e3, r, 1) , (e1, p, 1)) ,{

u1

〈0.3,0.7〉 ,u2

〈0.1,0.7〉 ,u3

〈0.3,1〉

}),(

((e3, r, 1) , (e3, p, 1)) ,{

u1

〈0.3,0.7〉 ,u2

〈0.3,0.7〉 ,u3

〈0.3,0.7〉

}),(

((e3, r, 1) , (e2, p, 0)) ,{

u1

〈0.3,0.7〉 ,u2

〈0.1,0.7〉 ,u3

〈0,0.6〉

}),(

((e2, p, 0) , (e1, p, 1)) ,{

u1

〈0.4,0.6〉 ,u2

〈0.6,0.8〉 ,u3

〈0,1〉

}),(

((e2, p, 0) , (e3, p, 1)) ,{

u1

〈0.4,0.5〉 ,u2

〈0.3,0.8〉 ,u3

〈0,0.7〉

}),

(((e2, p, 0) , (e2, p, 0)) ,

{u1

〈0.4,0.7〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉

})}.

Definition 3.2. If (F,A) and (G,B) are two vague soft expert sets over Uthen “(F,A) OR (G,B)" denoted by (F,A) ∨ (G,B) is defined by

(F,A) ∨ (G,B) = (H,A×B)

where H (α, β) = F (α)∪G(β) ∀α, β ∈ A×B and ∪ the vague soft expert union.



Example 3.2. Consider Example 3.1. Let

A = {(e1, p, 1), (e3, p, 1), (e3, r, 1), (e2, p, 0)} and B = {(e1, p, 1), (e3, p, 1), (e2, p, 0)}.

Suppose (F,A) and (G,B) are two vague soft expert sets over U such that

(F,A) =

{((e1, p, 1) ,

{u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉

}),(

(e3, p, 1) ,{

u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0.5〉

}),(

(e3, r, 1) ,{

u1

〈0.3,0.7〉 ,u2

〈0.1,0.3〉 ,u3

〈0.3,0.6〉

}),(

(e2, p, 0) ,{

u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉

})},

and

(G,B) =

{((e1, p, 1) ,

{u1

〈0.4,0.6〉 ,u2

〈0.7,0.7〉 ,u3

〈1,1〉

}),(

(e3, p, 1) ,{

u1

〈0.4,0.4〉 ,u2

〈0.3,0.7〉 ,u3

〈0.4,0.7〉

}),(

(e2, p, 0) ,{

u1

〈0.4,0.7〉 ,u2

〈0.6,0.7〉 ,u3

〈0,0〉

})}.

Then (F,A)∨

(G,B) = (H,A×B) ={(((e1, p, 1) , (e1, p, 1)) ,

{u1

〈0.5,0.5〉 ,u2

〈0.7,0.8〉 ,u3

〈1,0〉

}),

(((e1, p, 1) , (e3, p, 1)) ,

{u1

〈0.5,0.4〉 ,u2

〈0.6,0.7〉 ,u3

〈0.4,0〉

}),(

((e1, p, 1) , (e2, p, 0)) ,{

u1

〈0.4,0.7〉 ,u2

〈0.6,0.7〉 ,u3

〈0,0〉

}),(

((e3, p, 1) , (e1, p, 1)) ,{

u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0.5〉

}),(

((e3, p, 1) , (e3, p, 1)) ,{

u1

〈0.4,0.4〉 ,u2

〈0.6,0.6〉 ,u3

〈1,0.5〉

}),(

((e3, p, 1) , (e2, p, 0)) ,{

u1

〈0.4,0.4〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0〉

}),(

((e3, r, 1) , (e1, p, 1)) ,{

u1

〈0.4,0.6〉 ,u2

〈0.7,0.3〉 ,u3

〈1,0.6〉

}),(

((e3, r, 1) , (e3, p, 1)) ,{

u1

〈0.4,0.4〉 ,u2

〈0.3,0.3〉 ,u3

〈0.4,0.6〉

}),(

((e3, r, 1) , (e2, p, 0)) ,{

u1

〈0.4,0.7〉 ,u2

〈0.6,0.3〉 ,u3

〈0.3,0〉

}),(

((e2, p, 0) , (e1, p, 1)) ,{

u1

〈0.4,0.4〉 ,u2

〈0.7,0.3〉 ,u3

〈1,0.7〉

}),(

((e2, p, 0) , (e3, p, 1)) ,{

u1

〈0.4,0.4〉 ,u2

〈0.6,0.7〉 ,u3

〈0.4,0.7〉

}),

(((e2, p, 0) , (e3, p, 1)) ,

{u1

〈0.4,0.4〉 ,u2

〈0.7,0.3〉 ,u3

〈0.4,0〉

})}.



4. De Morgan’s Law on Vague Soft Expert Sets

In this section we give some De Morgan’s law on union and intersection ofvague soft expert sets and we prove the De Morgan’s law on vague soft expertsets. De Morgan’s law and its variants have been widely used in soft sets suchas by Adam and Hassan (2014b,e) on Q-fuzzy soft sets, multi Q-fuzzy soft sets(Adam and Hassan (2015)), and neutrosophic vague soft expert set (Al-Quranand Hassan (2016)).

Proposition 4.1. If (F,A) and (G,B) are two vague soft expert sets over U,then

1. (( F,A) ∧ (G,B))c

= (F,A)c ∨ (G,B)

c

2. ((F,A) ∨ (G,B))c

= (F,A)c ∧ (G,B)

c

Proof. 1. From the right hand side, we have

(F,A)c ∨ (G,B)

c= ((F c, eA) ∨ (Gc, eB))

= (J, (eA×eB)), whereJ(eα, eβ) = F c(eα) ∪Gc(eβ).

= (J, e (A×B)) .

Here e(A×B) means not in A and not in B at the same time.Suppose that (F,A) ∧ (G,B) = (H,A×B) .Therefore, ((F,A) ∧ (G,B))

c= (H,A×B)

c= (Hc, e(A×B))

Suppose we take ∀ (α, β) ∈ (A×B). We therefore haveHc(eα, eβ) = U −H(α, β),

= U − [F (α)⋂G (β)]

= [U − F (α)]⋃

[U −G (β)]= [F c (eα)]

⋃[Gc (eβ)]

= J(eα, eβ)Hence ((F,A) ∧ (G,B))

c= (J, e(A×B)).

From the discussion above, we thus have((F,A) ∧ (G,B))

c= (F,A)

c ∨ (G,B)c.

2. From the right hand side, we have

(F,A)c ∧ (G,B)

c= ((F c, eA) ∧ (Gc, eB))

= (K, (eA×eB)), whereK(eα, eβ) = F c(eα)⋂Gc(eβ),

= (K, e (A×B)) .

Suppose that (F,A) ∨ (G,B) = (H,A×B) .Therefore, ((F,A) ∨ (G,B))

c= (H,A×B)

c= (Hc, e(A×B)).



Suppose we take ∀(α, β) ∈ (A×B). We therefore haveHc(eα, eβ) = U −H(α, β),

= U − [F (α)⋃G (β)]

= [U − F (α)]⋂

[U −G (β)]= [F c (eα)]

⋂[Gc (eβ)]

= K(eα, eβ).Hence ((F,A) ∨ (G,B))

c= (K, e(A×B)) .

From the discussion above, we thus have((F,A) ∨ (G,B))

c= (F,A)

c ∧ (G,B)c.

5. Some propositions on vague soft sets

Proposition 5.1. If (F,A) , (G,B) and (H,C) are three vague soft expert setsover U , then

1. (F,A) ∩ ((G,B) ∩ (H,C)) = ((F,A) ∩ (G,B)) ∩ (H,C) ,

2. (F,A) ∩ (F,A) = (F,A) .

Proof. 1. We need to prove that(F,A) ∩ ((G,B) ∩ (H,C)) = ((F,A) ∩ (G,B)) ∩ (H,C) .

By using the Definition 2.11, we have

((G,B) ∩ (H,C)) (ε) =

G (ε) , if ε ∈ B − CH (ε) , if ε ∈ C −Bmin (G (ε) ∩H (ε)), if ε ∈ B ∩ C.

We consider the case when ε ∈ B ∩C as the other cases are trivial. Thus

(G,B) ∩ (H,C) (ε) = (G (ε) ∩H (ε) , B ∪ C).

We also consider here the case when ε ∈ A as the other cases are trivial.Thus

(F,A) ∩ ((G,B) ∩ (H,C)) (ε) = (F (ε) ∩ (G (ε) ∩H (ε)) , A ∪ (B ∪ C))

= ((F (ε) ∩G (ε)) ∩H (ε) , (A ∪B) ∪ C)

= ((F,A) ∩ (G,B)) ∩ (H,C) (ε).

2. The proof is straightforward.



Proposition 5.2. If (F,A) , (G,B) and (H,C) are three vague soft expert setsover U , then

1. (F,A) ∪ ((G,B) ∩ (H,C)) = ((F,A) ∪ (G,B)) ∩ ((F,A) ∪ (H,C)) ,

2. (F,A) ∩ ((G,B) ∪ (H,C)) = ((F,A) ∩ (G,B)) ∪ ((F,A) ∩ (H,C)) .

Proof. 1. We need to prove that(F,A) ∪ ((G,B) ∩ (H,C)) = ((F,A) ∪ (G,B)) ∩ ((F,A) ∪ (H,C)) ,

By using the Definitions 2.10 and 2.11, we have

((G,B) ∩ (H,C)) (ε) =

G (ε) , if ε ∈ B − CH (ε) , if ε ∈ C −Bmax (G (ε) ∩H (ε)) , if ε ∈ B ∩ C.

We consider the case when ε ∈ B ∩C as the other cases are trivial, thenwe have

(G,B) ∩ (H,C) (ε) = (G (ε) ∩H (ε) , B ∪ C).

We also consider here the case when ε ∈ A as the other cases are trivial,then we have

(F,A) ∪ ((G,B) ∩ (H,C)) (ε) = (F (ε) ∪ (G (ε) ∩H (ε)) , A ∪ (B ∪ C))

= ((F (ε) ∪G (ε)) , A ∪B) ∩ (F (ε) ∪H (ε) , A ∪ C)

= ((F,A) ∪ (G,B)) ∩ ((F,A) ∪ (H,C)) (ε).

2. We want to prove that(F,A) ∩ ((G,B) ∪ (H,C)) = ((F,A) ∩ (G,B)) ∪ ((F,A) ∩ (H,C)) ,

By using the Definitions 2.10 and 2.11, we have

((G,B) ∪ (H,C)) (ε) =

G (ε) , if ε ∈ B − CH (ε) , if ε ∈ C −Bmin (G (ε) ∪H (ε)) , if ε ∈ B ∩ C.

We consider the case when ε ∈ B ∩C as the other cases are trivial, thenwe have

(G,B) ∪ (H,C) (ε) = (G (ε) ∪H (ε) , B ∪ C).

We also consider here the case when ε ∈ A as the other cases are trivial,then we have

(F,A) ∩ ((G,B) ∪ (H,C)) (ε) = (F (ε) ∩ (G (ε) ∪H (ε)) , A ∪ (B ∪ C))



= ((F (ε) ∩G (ε)) , A ∪B) ∪ (F (ε) ∩H (ε) , A ∪ C)

= ((F,A) ∩ (G,B)) ∪ ((F,A) ∩ (H,C)) (ε).

Proposition 5.3. Let (F,A), (G,B) and (H,C) be vague soft expert sets.Then

1. (F,A) ∨ ((G,B) ∧ (H,C)) = ((F,A) ∨ (G,B)) ∧ ((F,A) ∨ (H,C)) .

2. (F,A) ∧ ((G,B) ∨ (H,C)) = ((F,A) ∧ (G,B)) ∨ ((F,A) ∧ (H,C)) .

Proof. 1. For any α ∈ A, β ∈ B and γ ∈ C and by using Proposition 5.2(a) and Definitions 2.10 and 2.11, we have

(F,A) ∨ ((G,B) ∧ (H,C)) = (F,A) ∪ ((G,B) ∩ (H,C))= ((F,A) ∪ (G,B)) ∩ ((F,A) ∪ (H,C))

= ((F,A) ∨ (G,B)) ∧ ((F,A) ∨ (H,C)) .

Hence (F,A)∨ ((G,B) ∧ (H,C)) = ((F,A) ∨ (G,B))∧ ((F,A) ∨ (H,C)) .

2. For any α ∈ A, β ∈ B and γ ∈ C and by using Proposition 5.2 (b) andDefinitions 2.10 and 2.11, we have(F,A) ∧ ((G,B) ∨ (H,C)) = (F,A) ∩ ((G,B) ∪ (H,C))

= ((F,A) ∩ (G,B)) ∪ ((F,A) ∩ (H,C))

= ((F,A) ∧ (G,B)) ∨ ((F,A) ∧ (H,C)) .

Hence (F,A)∧ ((G,B) ∨ (H,C)) = ((F,A) ∧ (G,B))∨ ((F,A) ∧ (H,C)) .

6. An application of vague soft expert set indecision making

Maji and Roy (2002) applied the theory of soft sets to solve a decisionmaking problem using rough mathematics. In this section, we present an ap-plication of vague soft expert set theory in a decision making problem basedon the Definitions 2.7 and 2.8. The problem we consider is as below.

Assume that a company wants to fill a position. There are three candidateswho form the universe U = {u1, u2, u3}. The hiring committee considers aset of parameters, E = {e1, e2, e3}, where the parameters ei(i = 1, 2, 3) standfor “experience", “computer knowledge" and “speak fluently" respectively. Let



X = {p, q, r} be a set of experts (committee members). After a serious discus-sion the committee constructs the following vague soft expert set.

(F,Z) = {((e1, p, 1), { u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉}), ((e1, q, 1), { u1

〈0.3,0.6〉 ,u2

〈0.9,0.9〉 ,u3

〈1,1〉}),((e1, r, 1), { u1

〈0.4,0.8〉 ,u2

〈0.2,0.2〉 ,u3

〈0.9,0.9〉}), ((e2, p, 1), { u1

〈0.2,0.8〉 ,u2

〈0.4,0.7〉 ,u3

〈0.5,0.6〉}),((e2, q, 1), { u1

〈0.3,0.4〉 ,u2

〈0.4,0.5〉 ,u3

〈0.6,0.8〉}), ((e2, r, 1), { u1

〈0.1,0.1〉 ,u2

〈0.8,0.9〉 ,u3

〈0.3,0.4〉})},((e3, p, 1), { u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0.5〉}), ((e3, q, 1), { u1

〈0.1,0.2〉 ,u2

〈0.2,0.6〉 ,u3

〈0.4,0.6〉}),((e3, r, 1), { u1

〈0.7,0.3〉 ,u2

〈0.1,0.3〉 ,u3

〈0.3,0.6〉}), ((e1, p, 0), { u1

〈0.1,0.1〉 ,u2

〈0.8,0.9〉 ,u3

〈0.3,0.3〉}),((e1, q, 0), { u1

〈0.3,0.4〉 ,u2

〈0.4,0.5〉 ,u3

〈0.6,0.8〉}), ((e1, r, 0), { u1

〈0.3,0.7〉 ,u2

〈0.1,0.3〉 ,u3

〈0.3,0.6〉}),((e2, p, 0), { u1

〈0.5,0.5〉 ,u2

〈0.6,0.8〉 ,u3

〈0,0〉}), ((e2, q, 0), { u1

〈0.3,0.6〉 ,u2

〈0.9,0.9〉 ,u3

〈1,1〉}),((e2, r, 0), { u1

〈0.4,0.8〉 ,u2

〈0.2,0.2〉 ,u3

〈0.9,0.9〉}), ((e3, p, 0), { u1

〈0.3,0.6〉 ,u2

〈0.6,0.6〉 ,u3

〈0.5,0.5〉}),((e3, q, 0), { u1

〈0.1,0.2〉 ,u2

〈0.2,0.6〉 ,u3

〈0.4,0.6〉}), ((e3, r, 0), { u1

〈0.3,0.7〉 ,u2

〈0.1,0.3〉 ,u3

〈0.3,0.6〉}).

In Table 1 and Table 2 we present the agree-vague soft expert set anddisagree-vague soft expert set respectively.

Table 1: Agree-vague soft expert set

U u1 u2 u3(e1, p) 〈0.5, 0.5〉〈0.6, 0.8〉〈0, 0〉(e1, q) 〈0.3, 0.6〉〈0.9, 0.9〉〈1, 1〉(e1, r) 〈0.4, 0.8〉〈0.2, 0.2〉〈0.9, 0.9〉(e2, p) 〈0.2, 0.8〉〈0.4, 0.7〉〈0.5, 0.6〉(e2, q) 〈0.3, 0.4〉〈0.4, 0.5〉〈0.6, 0.8〉(e2, r) 〈0.1, 0.1〉〈0.8, 0.9〉〈0.3, 0.4〉(e3, p) 〈0.3, 0.6〉〈0.6, 0.6〉〈0.5, 0.5〉(e3, q) 〈0.1, 0.2〉〈0.2, 0.6〉〈0.4, 0.6〉(e3, r) 〈0.7, 0.3〉〈0.1, 0.3〉〈0.3, 0.6〉

cj =n∑

i=1

(uij − uij) c1 = −1.4 c2 = −1.3 c3 = −0.9



Table 2: Disagree-vague soft expert set

U u1 u2 u3(e1, p) 〈0.1, 0.1〉〈0.8, 0.9〉〈0.3, 0.3〉(e1, q) 〈0.3, 0.4〉〈0.4, 0.5〉〈0.6, 0.8〉(e1, r) 〈0.3, 0.7〉〈0.1, 0.3〉〈0.3, 0.6〉(e2, p) 〈0.5, 0.5〉〈0.6, 0.8〉〈0, 0〉(e2, q) 〈0.3, 0.6〉〈0.9, 0.9〉〈1, 1〉(e2, r) 〈0.4, 0.8〉〈0.2, 0.2〉〈0.9, 0.9〉(e3, p) 〈0.3, 0.6〉〈0.6, 0.6〉〈0.5, 0.5〉(e3, q) 〈0.1, 0.2〉〈0.2, 0.6〉〈0.4, 0.6〉(e3, r) 〈0.3, 0.7〉〈0.1, 0.3〉〈0.3, 0.6〉

kj =n∑

i=1

(uij − uij) k1 = −2 k2 = −1.2 k3 = −1

The following algorithm, modified from Alkhazaleh and Salleh (2011) may befollowed by the company to fill the position.

1. Input the vague soft expert set (F,Z).

2. Find an agree-vague soft expert set and a disagree-vague soft expert set.

3. Find cj =n∑

i=1

(uij − uij) for agree-vague soft expert set.

4. Find kj =n∑

i=1

(uij − uij) for disagree-vague soft expert set.

5. Find sj = cj − kj .

6. Find m, for which sm = maxsj . Then sm is the optimal choice object.If m has more than one value, then any one of them could be chosen bythe company using its option.

Now we use this algorithm to find the best choice for the company to fillthe position. From Table 1 and Table 2 we have the following Table 3:

Table 3: sj = cj − kj

i cj kj sj1 −1.4 −2 0.62 −1.3 −1.2 −0.13 −0.9 −1 0.1



Then max sj = s1, so the committee will choose candidate 1 for the job.

In the case of soft expert set, the data sets would have to be binary. The softexpert set would then be defined as follows.

(F,Z) = {((e1, p, 1), {u1, u3}), ((e1, q, 1), {u2, u3}), ((e1, r, 1), {u2, u3}),((e2, p, 1), {u3}), ((e2, q, 1), {u1, u2, u3}), ((e2, r, 1), {u1, u2, u3})},((e3, p, 1), {u2, u3}), ((e3, q, 1), {u1, u3}), ((e3, r, 1), {u1, u2, }),((e1, p, 0), {u1, u2, u3}), ((e1, q, 0), {u1, u2, u3}), ((e1, r, 0), {u2, u3}),((e2, p, 0), {u1, u2, u3}), ((e2, q, 0), {u1, u2, u3}), ((e2, r, 0), {u2, u3}),((e3, p, 0), {u1, u2, u3}), ((e3, q, 0), {u1, u3}), ((e3, r, 0), {u2, u3}).

The values of sj would be as in Table 4. The maximum value would be thatof s1 and s3 . Thus there is a certain hesitancy to choose a candidate to fillup the vacant position. We have thus illustrated that a decision can be clearly

Table 4: sj = cj − kj

i cj kj sj1 5 6 −12 6 8 −23 8 9 −1

reached using vague soft expert set compared to that of soft expert set withouthesitancy.

7. Conclusion

In this paper, the basic concepts of a vague soft expert set are reviewed,and some basic operations on vague soft expert set are given with examples onunion and intersection of vague soft expert sets. Operations of AND and ORon vague soft expert sets are illustrated and the concept of De Morgan’s law ofvague soft expert set is also proven. In addition, this new extension not onlyprovides a significant addition to existing theories for handling uncertainties,but also leads to potential areas of further research and pertinent applications.

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